Geometric properties of the Cassinian metric

TL;DR

This paper investigates the geometric and distortion properties of the Cassinian metric under Möbius transformations, explores convexity of metric balls, and examines their inclusion relations with other hyperbolic-type metrics.

Contribution

It provides a sharp distortion theorem for the Cassinian metric under Möbius transformations and discusses convexity and inclusion properties of its metric balls.

Findings

Established a sharp distortion property of the Cassinian metric under Möbius transformations.

Analyzed local convexity properties of Cassinian metric balls in specific domains.

Investigated inclusion relations between Cassinian metric balls and other hyperbolic-type metric balls.

Abstract

In this paper we prove a sharp distortion property of the Cassinian metric under M\"obius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in some specific subdomains of $\mathbb{R}^n$. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.